Solve (3sqrt{12}-2sqrt{1/3}+sqrt{48})div(2sqrt{3})

$$(3 \sqrt{ 12 } -2 \sqrt{ \frac{ 1 }{ 3 } } + \sqrt{ 48 } ) \div (2 \sqrt{ 3 } )$$

Evaluate

$$\frac{14}{3}\approx 4.666666667$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{3\times 2\sqrt{3}-2\sqrt{\frac{1}{3}}+\sqrt{48}}{2\sqrt{3}}$$

Multiply 3 and 2 to get 6.

$$\frac{6\sqrt{3}-2\sqrt{\frac{1}{3}}+\sqrt{48}}{2\sqrt{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{6\sqrt{3}-2\times \frac{\sqrt{1}}{\sqrt{3}}+\sqrt{48}}{2\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$\frac{6\sqrt{3}-2\times \frac{1}{\sqrt{3}}+\sqrt{48}}{2\sqrt{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{6\sqrt{3}-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{48}}{2\sqrt{3}}$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{3}-2\times \frac{\sqrt{3}}{3}+\sqrt{48}}{2\sqrt{3}}$$

Express -2\times \frac{\sqrt{3}}{3} as a single fraction.

$$\frac{6\sqrt{3}+\frac{-2\sqrt{3}}{3}+\sqrt{48}}{2\sqrt{3}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{6\sqrt{3}+\frac{-2\sqrt{3}}{3}+4\sqrt{3}}{2\sqrt{3}}$$

Combine 6\sqrt{3} and 4\sqrt{3} to get 10\sqrt{3}.

$$\frac{10\sqrt{3}+\frac{-2\sqrt{3}}{3}}{2\sqrt{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10\sqrt{3} times \frac{3}{3}.

$$\frac{\frac{3\times 10\sqrt{3}}{3}+\frac{-2\sqrt{3}}{3}}{2\sqrt{3}}$$

Since \frac{3\times 10\sqrt{3}}{3} and \frac{-2\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\times 10\sqrt{3}-2\sqrt{3}}{3}}{2\sqrt{3}}$$

Do the multiplications in 3\times 10\sqrt{3}-2\sqrt{3}.

$$\frac{\frac{30\sqrt{3}-2\sqrt{3}}{3}}{2\sqrt{3}}$$

Do the calculations in 30\sqrt{3}-2\sqrt{3}.

$$\frac{\frac{28\sqrt{3}}{3}}{2\sqrt{3}}$$

Express \frac{\frac{28\sqrt{3}}{3}}{2\sqrt{3}} as a single fraction.

$$\frac{28\sqrt{3}}{3\times 2\sqrt{3}}$$

Cancel out 2\sqrt{3} in both numerator and denominator.

$$\frac{14}{3}$$

Factor

$$\frac{2 \cdot 7}{3} = 4\frac{2}{3} = 4.666666666666667$$